a) \(\frac{x + 2004}{x + 2005} + \frac{x + 2005}{2006} < \frac{x + 2006}{2007} + \frac{x + 2007}{2008} \Rightarrow \left(\right. \frac{x + 2004}{2005} - 1 \left.\right) + \left(\right. \frac{x + 2005}{2006} - 1 \left.\right) < \left(\right. \frac{x + 2006}{2007} - 1 \left.\right) + \left(\right. \frac{x + 2007}{2008} - 1 \left.\right) \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} < \frac{x - 1}{2007} + \frac{x - 1}{2008} \Rightarrow \frac{x - 1}{2005} + \frac{x - 1}{2006} - \frac{x - 1}{2007} - \frac{x - 1}{2008} < 0\)

\(\Rightarrow \left(\right. x - 1 \left.\right) \left(\right. \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} \left.\right) < 0 \left(\right. a \left.\right)\)

Nhận thấy: \(\frac{1}{2005} > \frac{1}{2007} , \frac{1}{2006} > \frac{1}{2008} \Rightarrow \frac{1}{2005} - \frac{1}{2007} > 0 , \frac{1}{2006} - \frac{1}{2008} > 0 \Rightarrow \frac{1}{2005} + \frac{1}{2006} - \frac{1}{2007} - \frac{1}{2008} > 0\)

\(\left(\right. a \left.\right) \Rightarrow x - 1 < 0 \Leftrightarrow x < 1\)

Vậy S={x∣x<1}

a,6x+2​+3x+5​>5x+3​+2x+6​<=>(6x+2​+1)+(3x+5​+1)>(5x+3​+1)+(2x+6​+1)<=>6x+8​+3x+8​>5x+8​+2x+8​<=>5x+8​+2x+8​−6x+8​−2x+8​<0<=>(x+8)(51​+21​−61​−31​)<0

Mà: \(1 / 5 + 1 / 2 + 1 / 6 - 1 / 3 > 0\)

\(= > x + 8 < 0\)

\(< = > x < - 8\) 

\(\frac{x - 2}{1007} + \frac{x - 1}{1008} < \frac{2 x - 1}{2017} + \frac{2 x - 3}{2015} < = > \left(\right. \frac{x - 2}{1007} - 1 \left.\right) + \left(\right. \frac{x - 1}{1008} - 1 \left.\right) < \left(\right. \frac{2 x - 1}{2017} - 1 \left.\right) + \left(\right. \frac{2 x - 3}{2015} - 1 \left.\right) < = > \frac{x - 1009}{1007} + \frac{x - 1009}{1008} < \frac{2 x - 2018}{2017} + \frac{2 x - 2018}{2015} < = > \frac{x - 1009}{1007} + \frac{x - 1009}{1008} - \frac{2 \left(\right. x - 1009 \left.\right)}{2017} - \frac{2 \left(\right. x - 1009 \left.\right)}{2015} < 0 < = > \left(\right. x - 1009 \left.\right) \left(\right. \frac{1}{1007} + \frac{1}{1008} - \frac{2}{2017} - \frac{2}{2015} \left.\right) < 0\)

Mà: \(1 / 1006 + 1 / 1008 - 2 / 2017 - 2 / 2015 > 0\)

\(= > x - 1009 < 0\)

\(< = > x < 1009\)

a,6x+2​+3x+5​>5x+3​+2x+6​<=>(6x+2​+1)+(3x+5​+1)>(5x+3​+1)+(2x+6​+1)<=>6x+8​+3x+8​>5x+8​+2x+8​<=>5x+8​+2x+8​−6x+8​−2x+8​<0<=>(x+8)(51​+21​−61​−31​)<0

Mà: \(1 / 5 + 1 / 2 + 1 / 6 - 1 / 3 > 0\)

\(= > x + 8 < 0\)

\(< = > x < - 8\) 

\(\frac{x - 2}{1007} + \frac{x - 1}{1008} < \frac{2 x - 1}{2017} + \frac{2 x - 3}{2015} < = > \left(\right. \frac{x - 2}{1007} - 1 \left.\right) + \left(\right. \frac{x - 1}{1008} - 1 \left.\right) < \left(\right. \frac{2 x - 1}{2017} - 1 \left.\right) + \left(\right. \frac{2 x - 3}{2015} - 1 \left.\right) < = > \frac{x - 1009}{1007} + \frac{x - 1009}{1008} < \frac{2 x - 2018}{2017} + \frac{2 x - 2018}{2015} < = > \frac{x - 1009}{1007} + \frac{x - 1009}{1008} - \frac{2 \left(\right. x - 1009 \left.\right)}{2017} - \frac{2 \left(\right. x - 1009 \left.\right)}{2015} < 0 < = > \left(\right. x - 1009 \left.\right) \left(\right. \frac{1}{1007} + \frac{1}{1008} - \frac{2}{2017} - \frac{2}{2015} \left.\right) < 0\)

Mà: \(1 / 1006 + 1 / 1008 - 2 / 2017 - 2 / 2015 > 0\)

\(= > x - 1009 < 0\)

\(< = > x < 1009\)

a,6x+2​+3x+5​>5x+3​+2x+6​<=>(6x+2​+1)+(3x+5​+1)>(5x+3​+1)+(2x+6​+1)<=>6x+8​+3x+8​>5x+8​+2x+8​<=>5x+8​+2x+8​−6x+8​−2x+8​<0<=>(x+8)(51​+21​−61​−31​)<0

Mà: \(1 / 5 + 1 / 2 + 1 / 6 - 1 / 3 > 0\)

\(= > x + 8 < 0\)

\(< = > x < - 8\) 

\(\frac{x - 2}{1007} + \frac{x - 1}{1008} < \frac{2 x - 1}{2017} + \frac{2 x - 3}{2015} < = > \left(\right. \frac{x - 2}{1007} - 1 \left.\right) + \left(\right. \frac{x - 1}{1008} - 1 \left.\right) < \left(\right. \frac{2 x - 1}{2017} - 1 \left.\right) + \left(\right. \frac{2 x - 3}{2015} - 1 \left.\right) < = > \frac{x - 1009}{1007} + \frac{x - 1009}{1008} < \frac{2 x - 2018}{2017} + \frac{2 x - 2018}{2015} < = > \frac{x - 1009}{1007} + \frac{x - 1009}{1008} - \frac{2 \left(\right. x - 1009 \left.\right)}{2017} - \frac{2 \left(\right. x - 1009 \left.\right)}{2015} < 0 < = > \left(\right. x - 1009 \left.\right) \left(\right. \frac{1}{1007} + \frac{1}{1008} - \frac{2}{2017} - \frac{2}{2015} \left.\right) < 0\)

Mà: \(1 / 1006 + 1 / 1008 - 2 / 2017 - 2 / 2015 > 0\)

\(= > x - 1009 < 0\)

\(< = > x < 1009\)